Timeline for Symmetries of irregular simplices
Current License: CC BY-SA 3.0
7 events
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| Oct 13, 2017 at 18:42 | comment | added | Hao Chen | But all these are only for dimension 3 and are not direct reference for what I'm asking. | |
| Oct 13, 2017 at 18:41 | comment | added | Hao Chen | FYI, I consulted with the wikipedia editor. He used a software to calculate the subgroups of the tetrahedral group. He also mentioned Conway's book "the symmetries of things". There is a poset for the octahedral group, from which one can extract a poset for the tetrahedral group (if he manages to understand the notation system). | |
| Oct 7, 2017 at 9:39 | comment | added | Peter Heinig | [...] op. cit. has a general title, yet is not sufficient by itself to answer this question. The most difficult part of the question (to me) seems to rigorously show that the question even can be reduced to graph theory; it is not clear that all the relevant graph-theoretic symmetries can be realized by a Euclidean symmetry. It seems that if someone likes to get into this question, then carefully working out the connection between coloring-preserving automorphisms of complete graphs and simplex-realization-preserving Euclidean symmetries is the most important step towards an answer. | |
| Oct 7, 2017 at 9:35 | comment | added | Peter Heinig | [...] many such isomorphism groups, starting from the trivial group (which arises for most 'irregular' geometric realizations) all the way to the symmetric group $\mathrm{Sym}(n)$. The OP also points out that any such geometric realization defines an inclusion of groups of the relevant symmetryi group into the autormorphism group of a $k$-colored graph in the sense of e.g. [Mariusz Grech, Andrzej Kisielewicz: All totally symmetric colored graphs. arXiv:1201.4464v1]. The OP seems right in that it is not easy to read-off an answer to the question from the literature, in particular [...] | |
| Oct 7, 2017 at 9:27 | comment | added | Peter Heinig | As a help for others to find an answer: this seems an interesting and perfectly well-defined question; to paraphrase: the OP is asking for citable sources on what are all the possible isomorphism types of of the (necessarily finite) subgroups which arise as the set-wise stabiliser, inside the full Euclidean group $E(n)$, of a specified geometric realization of a simplicial complex in $\mathbb{R}^n$, the realization being arbitrary (except, perhaps, being piecewise-linear, but even that needn't be said, I think). There are indeed [...] | |
| Oct 7, 2017 at 0:23 | history | edited | Hao Chen |
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| Oct 7, 2017 at 0:09 | history | asked | Hao Chen | CC BY-SA 3.0 |